I'm currently reading the "Reflection groups and Coxeter groups" textbook by Humphreys, but I'm struggling to understand a paragraph at the beginning of section 3.9.
For a Weyl group $W$, it states "Since the trace of each $w \in W$ is real, it is clear that the only elements of $W$ having $n-1$ eigenvalues equal to $1$ are the identity and the $N$ reflections (where $N$ is the number of positive roots)".
The problem is, for me it isn't clear - can anyone explain it to me? I've been agonising over this for a stupid amount of time at this point (which might have something to do with it being $1$am...).
2026-02-22 17:04:03.1771779843
Eigenvalues of elements of a Weyl group
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